Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$ , and the remaining one has charge $x$ . The perpendicular from each charge to the nearest hexagon side passes through the center $O$ of the hexagon and is bisected by the side.

Which of the following statement(s) is(are) correct in SI units [2022]

Question

Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$ , and the remaining one has charge $x$ . The perpendicular from each charge to the nearest hexagon side passes through the center $O$ of the hexagon and is bisected by the side.

Which of the following statement(s) is(are) correct in SI units [2022]

Which of the following statement(s) is(are) correct in SI units? [2022]

Smart Achievers · Accepted Answer

(1, 2, 3)

(1) When $x = q$ , the situation is symmetric.

So, electric field at $O$ is zero.

$\Rightarrow (1) is correct.$

(2) When $x = - q$ , then $E_{0} = \frac{1}{4 π ε_{0}} \frac{q}{{(\sqrt{3} a)}^{2}} \times 2$

$\Rightarrow E_{0} = \frac{q}{6 π ε_{0} a^{2}}$

$\Rightarrow (2) is correct.$

(3) When $x = 2 q$ ,

$V_{0} = 5 \times \frac{K q}{\sqrt{3} a} + \frac{K (2 q)}{\sqrt{3} a}$ $\Rightarrow V_{0} = \frac{7 q}{4 \sqrt{3} π ε_{0} a}$

$\Rightarrow (3) is correct.$

(4) When $x = - 3 q$ ,

$V_{0} = 5 \times \frac{K q}{\sqrt{3} a} + \frac{K (- 3 q)}{\sqrt{3} a}$ $= \frac{1}{4 π ε_{0}} \frac{2 q}{\sqrt{3} a} = \frac{q}{2 \sqrt{3} π ε_{0} a}$

$\Rightarrow (4) is incorrect.$

Solution

1 When $x = q$ , the magnitude of the electric field at $O$ is zero.

2 When $x = - q$ , the magnitude of the electric field at $O$ is $\frac{q}{6 π ε_{0} a^{2}} .$

3 When $x = 2 q$ , the potential at $O$ is $\frac{7 q}{4 \sqrt{3} π ε_{0} a} .$

4 When $x = - 3 q$ , the potential at $O$ is $- \frac{3 q}{4 \sqrt{3} π ε_{0} a} .$

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Solution

1 When x=q, the magnitude of the electric field at O is zero.

2 When x=-q, the magnitude of the electric field at O is q6πε0a2.

3 When x=2q, the potential at O is 7q43πε0a.

4 When x=-3q, the potential at O is -3q43πε0a.

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1 When $x = q$ , the magnitude of the electric field at $O$ is zero.

2 When $x = - q$ , the magnitude of the electric field at $O$ is $\frac{q}{6 π ε_{0} a^{2}} .$

3 When $x = 2 q$ , the potential at $O$ is $\frac{7 q}{4 \sqrt{3} π ε_{0} a} .$

4 When $x = - 3 q$ , the potential at $O$ is $- \frac{3 q}{4 \sqrt{3} π ε_{0} a} .$